Chaos, Fractals, and Arcadia

Thomasina's fern

How did Thomasina produce an algorithm that yields a natural form? We
will illustrate this with the simplest case, a fern. A leaf is a
little more difficult and a little less spectacular than a fern.
We need to
describe an algorithm that, when iterated as in the chaos game,
yields an image of a fern. The fern we will produce is often called
the Barnsley fern after the mathematician who popularized this
procedure [B].

To do this we start with a square as before. We will describe four
linear contractions on this square. Unlike the previous two examples,
these contractions will involve more than just simple dilations; they
will involve rotations and flips as well. Here is the first
operation: squeeze and distort the square linearly so that its image
appears as in Figure 5A. Note that the square is compressed from the
bottom and from both sides, and then rotated a little.
Figure 5B displays the effects of the next two contractions.
The left hand rectangle is obtained by first shrinking the square
into a rectangle,
then shearing and rotating to the left. The second is obtained in
similar fashion, except that the square is first flipped along its
vertical axis, and then contracted, sheared, and rotated to the
right.
features more contraction from the top than from the bottom, then
rotation in opposite directions. The right hand rectangle is then
flipped along In Figure 5C we see the final
contraction: the entire square is crushed to a line segment in the
horizontal direction, then compressed again in the short
direction
to yield the short vertical line segment indicate.

Fig. 5. The four contractions for the Barnsley fern. In each case,
the entire grey square is contracted linearly into the four black rectangles.

Each of these rules can be described concisely using some matrix
algebra. In the appendix, we give exact formulas for each of these
transformations as well as a brief discussion of where they come from.

Now we play the chaos game with these rules as the four constituent
moves. However, instead of randomly choosing a particular
contraction, we will choose the rules to apply with differing
probability.
We will apply the first contraction with the highest probability,
namely 85 % of the time. Contractions 2 and 3 will be invoked
with probability .07, and the final contraction will be called with
probability only .01. When this algorithm is carried out using a
computer, the dots slowly fill the screen to reveal a lifelike image
of a fern.

As Valentine explains to Hannah, "If you knew the
algorithm and fed it back say ten thousand times, each time there'd be
a dot somewhere on the screen. You'd never know where to expect the
next dot. But gradually you'd start to see this shape, because every
dot would be inside this shape of a leaf (or a fern). It
wouldn't be a leaf, it would be a mathematical object. But yes. The
unpredictable and the predetermined unfold together to make everything
the way it is. It's how nature creates itself, on every scale, the
snowflake and the snowstorm."

To see the fern leaf unfold gradually on the screen, click
here. You will need a QuickTime
viewer to see this animation.

In case you don't have the means to view the animation, here are a few
snapshots of the fern unfolding.
Incidentally, all of these images were computed
using the software package Fractal Attraction [LC]

Fig. 6. The results of the chaos game played with
5,000, 10,000, and 50,000, iterations.